Abstract We study the mathematical allegory in Edwin Abbott’s novella Flatland: A Romance of Many Dimensions, whose ideas influenced mathematics, science fiction, education, philosophy, cosmology, modern art, and computer graphics, among other areas. Abbott, a schoolmaster and biblical, English literature, and Classics scholar, used n-dimensional geometry, up to the fourth dimension, to demonstrate how it is possible to understand an idea or object that we cannot construct or perceive. We provide an overview of nineteenth-century geometry, leading up to the discovery of different geometries and subsequently, the exploration of higher dimensions. We demonstrate the dimensional analogy, a mathematical exercise that introduces the fourth dimension—Abbott’s mathematical allegory. We explore Flatland’s various dimensions: as an introduction to higher-dimensional geometry, as a model of Plato’s cave, and as satire of Victorian hierarchical society and its treatment of women, and we provide a brief survey of novels and films Flatland has inspired.

We give an algorithm for computing inseparable endomorphisms of a supersingular elliptic curve $E$ defined over $\mathbb F_{p^2}$, which, conditional on GRH, runs in expected $O(\sqrt{p}(\log p)^2(\log\log p)^3)$ time. With two calls to this algorithm, we compute a Bass suborder of $\text{End}(E)$, improving on the results of Eisenträger, Hallgren, Leonardi, Morrison, and Park (ANTSXIV) who only gave a heuristic algorithm for computing a Bass suborder. We further improve on the results of Eisenträger et al. by removing the heuristics involved in an algorithm for recovering $\text{End}(E)$ from a Bass suborder. We conclude with an argument that $O(1)$ endomorphisms generated by our algorithm along with negligible overhead suffice to compute $\text{End}(E)$, conditional on a heuristic assumption about the distribution of the discriminants of these endomorphisms.

Let σ be the usual sum-of-divisors function. We say that a and b form a harmonious pair if [Formula: see text]; equivalently, the harmonic mean of [Formula: see text] and [Formula: see text] is 2. For example, 4 and 12 form a harmonious pair, since [Formula: see text] and [Formula: see text]. Every amicable pair is harmonious, but there are many others. We show that the count of numbers that belong to a harmonious pair having both members in [1, x] is at most [Formula: see text], as x → ∞.

In this paper, we show that there are infinitely many Sierpinski numbers in the sequence of triangular numbers, hexagonal numbers, and pentagonal numbers. We also show that there are infinitely many Riesel numbers in the same sequences. Furthermore, we show that there are infinitely many n-gonal numbers that are simultaneously Sierpinski and Riesel.

As an illustration of the use of Corollary 6 consider the result of [3], where it was shown that a smallest group with a nontrivial odd-order automorphism group has order 36. To eliminate the possibility of there being groups of smaller order satisfying the hypothesis, part of the proof in [3] considered individually groups of orders pq, p2q, p2q2, pqr and showed each of these had an automorphism group of even order. This fact follows immediately from the Corollary.

We address conjectures of P. Erdős and conjectures of Y.-G. Chen concerning the numbers in the title. We obtain a variety of related results, including a new smallest positive integer that is simultaneously a Sierpiński number and a Riesel number and a proof that for every positive integer r, there is an integer k such that the numbers k,k2,k3,…,kr are simultaneously Sierpiński numbers.